More Framesand-Arrows Problems
Objective To introduce Frames-and-Arrows problems in
which the “arrow rule” is missing.
w
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ePresentations
eToolkit
Algorithms
Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Count forward and backward by 1s, 2s, 3s,
5s, and 10s from a given number. [Number and Numeration Goal 1]
• Find the missing numbers in a Framesand-Arrows problem given the rule. [Patterns, Functions, and Algebra Goal 1]
• Identify rules in Frames-and-Arrows
problems. [Patterns, Functions, and Algebra Goal 1]
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Practicing Adding on the
Number Grid
Math Masters, p. 76
Children practice adding on a
number grid.
Math Boxes 3 9
Math Journal 1, p. 44
Children practice and maintain skills
through Math Box problems.
Home Link 3 9
• Create Frames-and-Arrows problems. [Patterns, Functions, and Algebra Goal 1]
Key Activities
Math Masters, p. 77
Children practice and maintain skills
through Home Link activities.
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Finding Patterns on the Number Line
Math Masters, p. 78; p. 312 (optional)
Children find patterns as they count on a
number line.
ENRICHMENT
Making Frames-and-Arrows Diagrams
from Skip-Counting Patterns
Math Masters, p. 79
Children skip count on a number grid and
apply those numbers to Frames-and-Arrows.
Children complete Frames-and-Arrows
diagrams in which a sequence of numbers
is given and they are to find the rule.
Ongoing Assessment:
Informing Instruction See page 225.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 43. [Patterns, Functions, and Algebra Goal 1]
Materials
Math Journal 1, p. 43 and inside back cover
Home Link 38
Math Masters, pp. 336A and 336B
transparency of Math Masters, p. 312
(optional)
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Unit 3
Visual Patterns, Number Patterns, and Counting
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Getting Started
Mental Math and Reflexes
Home Link 3 8 Follow-Up
Play Ten-Frame Top-It. See Lesson 2-3 for detailed
instructions. Depending on children’s comfort level with
ten frames, you may wish to have children write the addition
facts for their cards. For example, if a player’s ten-frame card
shows 7, the player would write 7 + 3 = 10 or 3 + 7 = 10.
Briefly go over the answers.
1 Teaching the Lesson
Finding the Arrow Rule
WHOLE-CLASS
ACTIVITY
Suggestions:
(Math Masters, p. 312)
Algebraic Thinking On the board, write a set of numbers that
represents a simple skip count, such as 5, 10, 15, and 20. Ask
children to be detectives and determine how many hops it takes to
move from one number to the next. 5 Tell children that today they
will be solving mystery problems like this one.
Begin by doing a few problems like those in Lesson 3-8, in which
the rule and the first few numbers are given. You can use the
suggestions given in the margin. (Write the problems on the
board, or use a transparency of Math Masters, page 312.)
When you feel that children are ready, pose the following problem:
Rule
?
1
3
5
7
9
Ask children to guess the rule. Answers include “Count up by 2s,”
“Add 2,” “2 more,” and “+ 2.” Encourage children to suggest
various ways to state the rule.
Continue with other problems in which the rule is missing. (See
the second set of suggestions in the margin.) Children should use
the number grid on the inside back cover of their journals if they
need help figuring out the rule or pattern.
Rules
Watch for children who have difficulty stating the rule. Encourage them to
determine how many hops there are between two numbers by using the number
grid. Record the number of hops in the rule box. Then help them determine if
they are hopping up or back. Show them how to record this information with a
+ or - sign in the rule box.
Missing
Frames
7, 9
11, 13, 15
12, 10
8, 6, 4
4, 7
10, 13, 16
60, 50
40, 30, 20
Count up by 2s
Count back by 2s
Add 3
- 10
NOTE Once children have determined that
the numbers in the frames increase from left
to right (indicating that you should add), you
may wish to explain that finding the arrow
rule is like finding the unknown number that
makes these number models true.
1+
=3
3+
=5
5+
=7
7+
=9
Suggestions:
Frames
Ongoing Assessment: Informing Instruction
First
Frames
Missing Rules
2, 4, 6, 8, 10
Count up by 2s, or add 2
15, 14, 13, 12, 11
Count back by 1s, or subtract 1
3, 6, 9, 12, 15
Count up by 3s, or add 3
15, 13, 11, 9, 7
Count back by 2s, or subtract 2
20, 30, 40, 50, 60
Count up by 10s, or add 10
Lesson 39
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Student Page
Making up Frames-and-
WHOLE-CLASS
ACTIVITY
Arrows Problems
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
LV
VIN
IIN
NG
(Math Masters, p. 312)
Algebraic Thinking Ask children to make up their own Framesand-Arrows problems. Since the simplest problems are those in
which the rule and the first number are given, begin with these.
Then encourage children to suggest more difficult problems.
Adjusting the Activity
Suggest that children make up problems in which the first frame is
empty. This requires counting back to fill in the first frame.
A U D I T O R Y
Math Journal 1, p. 43
K I N E S T H E T I C
T A C T I L E
Solving Frames-and-
V I S U A L
PARTNER
ACTIVITY
Arrows Problems
PROBLEM
PRO
P
R
ROB
RO
ROBLEM
OB
BLE
BL
LE
L
EM
SO
S
SOLVING
OL
O
LV
VIN
IIN
NG
(Math Journal 1, p. 43)
NOTE Math Journal 1, page 43, will be used
again for the optional Enrichment activity in
Lesson 3-10.
Algebraic Thinking Partners work together to solve the problems.
Encourage children to help each other and to check each other’s
work. They should use a number grid to help them with the more
difficult problems.
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 43
Problem 1
Use journal page 43, Problem 1 to assess children’s understanding of Frames
and Arrows. Children are making adequate progress if they are able to
correctly answer Problem 1. Some children may be able to correctly answer
more problems.
[Patterns, Functions and Algebra Goal 1]
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Unit 3 Visual Patterns, Number Patterns, and Counting
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Teaching Master
2 Ongoing Learning & Practice
Name
Date
LESSON
Adding on the Number Grid
39
0
1
Practicing Adding on the
INDEPENDENT
ACTIVITY
2
3
4
5
6
7
8
9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
Number Grid
31 32 33 34 35 36 37 38 39 40
(Math Masters, p. 76)
51 52 53 54 55 56 57 58 59 60
41 42 43 44 45 46 47 48 49 50
61 62 63 64 65 66 67 68 69 70
Use Math Masters, page 76 to provide additional practice adding
on the number grid.
1. Start at 25. Count up 3. Where do you end up?
28
25 + 3 =
28
2. Start at 19. Count up 6. Where do you end up?
25
19 + 6 =
25
py g
Math Boxes 3 9
INDEPENDENT
ACTIVITY
3. Start at 38. Count up 2. Where do you end up?
38 + 2 =
40
g
40
(Math Journal 1, p. 44)
p
4. Start at 57. Count up 10. Where do you end up?
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 3-11. The skills in Problem 4
preview Unit 4 content.
67
57 + 10 =
67
Try This
49
5. 29 + 20 =
40
6. 25 + 15 =
Math Masters, p. 76
Home Link 3 9
INDEPENDENT
ACTIVITY
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(Math Masters, p. 77)
Home Connection Children find the missing rules in
Frames-and-Arrows problems.
Home Link Master
Name
Student Page
Date
HOME LINK
39
Date
Find the Rule
LESSON
39
Family
Note
Today we worked with Frames-and-Arrows diagrams in which the rule was missing. You may
want to refer back to the Family Note for Lesson 3-8 and review the Frames-and-Arrows
routine.
1. Use your number line.
2. Record the time.
Start at 0.
Please return this Home Link to school tomorrow.
11
10
Count up 6.
Show someone at home how to find the rules.
Then write each rule.
1.
Math Boxes
6
You end at
Sample answers given for Problems 1–3.
.
3
4
7
3
5
7
9
2
8
Rule
Add 2
6
5
11
7
half-past
2.
Rule
3.
Add 5
1
9
6
0+6=
12
5
10
15
20
25
18
15
12
9
6
Rule
20
15
Count by 5s
25
30
o’clock
35
3.
Subtract 3
4. Fill in the blanks.
10
g
g
Rule
60
Practice
,
20
,
70
,
30
,
80
,
40
,
90
,
50
,
100
,
4. Circle the winning card in Top-It.
18
12
77
Math Masters, p. 77
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Math Journal 1, p. 44
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Lesson 39
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Teaching Master
3 Differentiation Options
READINESS
Finding Patterns on the
INDEPENDENT
ACTIVITY
5–15 Min
Number Line
(Math Masters, p. 78)
Math Masters, p. 78
Algebraic Thinking To provide children with a visual model for
how to find the rule in Frames-and-Arrows sequences, have
children count on a number line. Children complete Math Masters,
page 78. When children have finished the page, share ideas on how
to solve the problems and discuss which problem was the most
difficult. Consider having children state rules for each problem.
The rules could be to either add or subtract, depending on which
direction children choose to move. Consider recording the problems
as Frames-and-Arrows diagrams on Math Masters, page 312. Have
children read the number of hops for each number line. For
example, “There were three hops between the numbers on this
number line.”
ENRICHMENT
Making Frames-and-Arrows
INDEPENDENT
ACTIVITY
5–15 Min
Diagrams from Skip-Counting
Patterns
(Math Masters, p. 79)
Teaching Master
Algebraic Thinking To further explore Frames-and-Arrows
diagrams, have children complete Frames-and-Arrows diagrams
for patterns on the number grid. Have children fill in each
number-grid section on Math Masters, page 79 with a series of
consecutive numbers. Invite them to shade a skip-counting pattern
on the number-grid section. Challenge them to transfer their skip
counting pattern into a Frames-and-Arrows diagram. Encourage
children to repeat the process with a different section of the
number grid.
Math Masters, p. 79
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